Next, we examine the energy dependence of
the LDOS in the vortex states on the honeycomb lattice. The LDOS is
defined as
\(N(\mathbf {R}_{i},E)=N_{\uparrow }(\mathbf {R}_{i},E)+N_{\downarrow }(\mathbf {R}_{i},E)\) 
with
\(N_{\uparrow }(\mathbf {R}_{i},E)=-\sum _{n}|u_{A(B),n,i,\uparrow }|^{2}f^{\prime }(E_{n}-E)\) 
and
\(N_{\downarrow }(\mathbf {R}_{i},E)=-|v_{A(B),n,i,\downarrow }|^{2}f^{\prime }(E_{n}+E)\) 
being the spin-resolved LDOS for spin-up and spin-down states,
respectively. In order to reduce the finite size effect, the
calculations of the LDOS are carried out on a periodic lattice which
consists of \(16\times 8\)  parallelogram vortex unit cells, with each
vortex unit cell being the size of
\(24\mathbf {a_{1}}\times 48\mathbf {a_{2}}\) . In Fig. REF , we
plot a series of the spin-resolved LDOS as a function of energy at
sites along the zigzag direction moving away from the vortex center
for \(\beta =0.0\)  and \(\beta =0.04\) , respectively. For comparison, we
have also displayed the LDOS at the midpoint between the two NN
vortices, which resembles the U-shaped full gap feature for the bulk
system. In the absence of ISOC, the states of spin-up and spin-down
are nearly equal occupation and empty in the vortex core, as shown
in Fig. REF (a), in accordance with the empty cores without
the induced magnetic orders. Besides the almost identical LDOS line
shapes for the spin-up and spin-down states, the LDOS shown in
Fig. REF (a) exhibits another two prominent features within
the SC gap edges. On one hand, the LDOS shows the pronounced
discrete energy peaks inside the core region with one located near
the zero energy and others located at finite energies, as indicated
by the dashed vertical lines in the figure. Here, the asymmetric
line shape of the LDOS with respect to zero energy reflects the lack
of particle-hole symmetry as the chemical potential \(\mu \)  deviates
from zero for the filling factor \(n\)  being greater than the half
filling (\(n>1\) ). Due to the particle-hole asymmetry, the
finite-energy bound states at the core site only appear on the \(E>0\) 
side [1]} (There are also weak peaks at finite energies on
the \(E<0\)  side when moving away from the core center.). The
existence of the zero-energy vortex core sates in the Dirac fermion
system have been predicted analytically by Jackiw and Rossi in terms
of the zero-energy solutions of relativistic field
theory [2]}. Although these zero-energy solutions were
subsequently demonstrated that the existence of these zero-energy
solutions is connected to an index theorem [3]} and the zero
modes were shown to exist in the Dirac continuum theory of the
honeycomb lattice at half filling [4]}, the zero-energy
levels split when adopting a honeycomb lattice model description by
setting the size of the vortex core to be zero [5]}. It is
also found that the energy splitting decreases with the vortex size
and leads to the near-zero-energy states in the circumstance of
finite core size [5]}. While the notion of the zero-energy
vortex core states presents an important subject of study being
worthy of further research, we identify the near-zero-energy vortex
core states here in a self-consistent manner by employing the
honeycomb lattice model, where the band structure has the Dirac-type
dispersion near the half filling. On the other hand, though the
peaks' intensities are suppressed as the site departing from the
core center, the energy levels of these peaks are almost independent
of positions. It is worth while to note that a dispersionless
zero-energy conductance peak has been recently observed inside the
SC vortex core by Chen's group [7]} in the kagome
superconductor CsV\(_{3}\) Sb\(_{5}\) , which shares the lattice structure
with component of hexagonal honeycomb and the electronic structure
with Dirac points in a manner similar to those in honeycomb
lattices. How the calculated results with near-zero-energy peaks in
the present study relate to the experimental observations, and
whether these near-zero-energy vortex core states have a common
underlying symmetrical cause, constituting another fascinating
questions deserving further studies.
